TY  - CHAP
A1  - Burgeth, Bernhard
A1  - Kleefeld, Andreas
A1  - Zhang, Eugene
A1  - Zhang, Yue
ED  - Baudrier, Étienne
ED  - Naegel, Benoît
ED  - Krähenbühl, Adrien
ED  - Tajine, Mohamed
T1  - Towards Topological Analysis of Non-symmetric Tensor Fields via Complexification
T2  - Discrete Geometry and Mathematical Morphology
N2  - Fields of asymmetric tensors play an important role in many applications such as medical imaging (diffusion tensor magnetic resonance imaging), physics, and civil engineering (for example Cauchy-Green-deformation tensor, strain tensor with local rotations, etc.). However, such asymmetric tensors are usually symmetrized and then further processed. Using this procedure results in a loss of information. A new method for the processing of asymmetric tensor fields is proposed restricting our attention to tensors of second-order given by a 2x2 array or matrix with real entries. This is achieved by a transformation resulting in Hermitian matrices that have an eigendecomposition similar to symmetric matrices. With this new idea numerical results for real-world data arising from a deformation of an object by external forces are given. It is shown that the asymmetric part indeed contains valuable information.
Y1  - 2022
SN  - 978-3-031-19897-7
U6  - https://doi.org/10.1007/978-3-031-19897-7_5
N1  - Second International Joint Conference, DGMM 2022, Strasbourg, France, October 24–27, 2022
N1  - Corresponding author: Andreas Kleefeld
SP  - 48
EP  - 59
PB  - Springer
CY  - Cham
ER  - 
TY  - JOUR
A1  - Harris, Isaac
A1  - Kleefeld, Andreas
T1  - Analysis and computation of the transmission eigenvalues with a conductive boundary condition
JF  - Applicable Analysis
N2  - We provide a new analytical and computational study of the transmission eigenvalues with a conductive boundary condition. These eigenvalues are derived from the scalar inverse scattering problem for an inhomogeneous material with a conductive boundary condition. The goal is to study how these eigenvalues depend on the material parameters in order to estimate the refractive index. The analytical questions we study are: deriving Faber–Krahn type lower bounds, the discreteness and limiting behavior of the transmission eigenvalues as the conductivity tends to infinity for a sign changing contrast. We also provide a numerical study of a new boundary integral equation for computing the eigenvalues. Lastly, using the limiting behavior we will numerically estimate the refractive index from the eigenvalues provided the conductivity is sufficiently large but unknown.
KW  - Boundary integral equations
KW  - Inverse spectral problem
KW  - Conductive boundary condition
KW  - Transmission eigenvalues
Y1  - 2020
U6  - https://doi.org/10.1080/00036811.2020.1789598
SN  - 1563-504X
VL  - 101
IS  - 6
SP  - 1880
EP  - 1895
PB  - Taylor & Francis
CY  - London
ER  -